Fegan, Howard D.; Gilkey, Peter Invariants of the heat equation. (English) Zbl 0584.58041 Pac. J. Math. 117, 233-254 (1985). Let M be a compact Riemannian manifold without boundary and \(P: C^{\infty}(V)\to C^{\infty}(V)\) be a self-adjoint elliptic differential operator with leading symbol p(x,\(\xi)\), a matrix for \(x\in M\) and \(\xi \in T^*M_ x\). The authors demonstrate that if p(x,\(\xi)\) is positive definite then the set \(\{e_ n(x,\exp (-tP))\}\), which are obtained by expanding the kernel K(x,x) of the operator exp(-tP) when \(t>0\), span all the invariants arising from the heat equation Tr exp(- tP). In particular they assert that no new invariant could be obtained from \(e_ n(x,A(P)\exp (-tB(P))\), where A(r) and B(r) are constant coefficient polynomials. In the case that p(x,\(\xi)\) is indefinite the set \(\{e_ n(x,\exp (-tP^ 2),e_ n(x,P \exp (-tP^ 2))\}\) span all invariants. The authors emphasize the importance of studying the invariant of heat equation by pointing out that this problem is related to a number of profound theorems such as the Atiyah-Singer index theorem, the Gauss-Bonnet theorem. The authors also show that the above-mentioned invariants are closely related to the study of zeta and eta functions. Reviewer: G.Tu Cited in 10 Documents MSC: 58J35 Heat and other parabolic equation methods for PDEs on manifolds 53C20 Global Riemannian geometry, including pinching 47A10 Spectrum, resolvent Keywords:invariant; self-adjoint elliptic differential operator; Atiyah-Singer index theorem; Gauss-Bonnet theorem; zeta and eta functions PDFBibTeX XMLCite \textit{H. D. Fegan} and \textit{P. Gilkey}, Pac. J. Math. 117, 233--254 (1985; Zbl 0584.58041) Full Text: DOI